The song did indeed cross over the pond. As Susan of DT mentioned in 2000, the song was written by Dan Emmett and started life as "Jordan Am a Hard Road to Travel."

Emmett wrote the song in 1853, so broadsides existing around then would make sense.

This is (partly) backed up by an article published in 1857 in The Birmingham Daily Post, Dec. 14th, 1857, p. 3, which writes:

"During each Entertainment will be pourtrayed [sic], through the Medium of Glees, Songs, Overtures, Dances, Lectures, Refrains, Sayings and Doings, the Oddities, Peculiarities, Comicalities, Eccentricities, and Whimsicalities, of the Slaves and Free Blacks of America.

"The Programme will be changes each Evening, and consist of the following SONGS, &c. :--.
...
"T'OTHER SIDE OF JORDAN"

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T'other Side of Jordan or The Other Side of Jordan is pretty clearly just a derivative work of Jordan Am A Hard Road to Travel. A copy from 1853 can be found in the Levy Sheet Music Collection; it attributed the song to Luke West. An 1854 copy in the Library of Congress cited J. R. Thomas as the arranger and attributed the song to Christy's Minstrels, a band with which, notably, West sang with in 1853.

Though it's hard to say which came first, I think "Jordan Am A Hard Road" came before "T'Other Side of Jordan" simply because I found references to the sheet music being for sale sooner, in addition to the belief of most historians that Emmett wrote "Jordan Am A Hard Road" (though I didn't find any explicit mentions of "T'Other Side of Jordan").

"Jordan Am A Hard Road" could be found for sale in December 1852 (source), though the earliest attribution of the song to Emmett I found was 1854.

"T'Other Side of Jordan" wasn't found for sale until December 1853 (source)